Problem: The following system of equations is represented by the matrix equation $\text{A}\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right]=\vec{b}$. $\begin{aligned}12x+15y+2z&=19 \\x-8y+6z&=14 \\15x+12y-6z&=3\end{aligned}$ ${A}=$ $\vec{b}=$ Represent each row and column in the order in which the variables and equations appear.
Explanation: The Strategy A system of equations can be represented by a matrix equation $\text{A}\vec{x}=\vec{b}$, where $\text{A}$ is the coefficient matrix, $\vec{x}$ is the variables vector, and $\vec{b}$ is the constants vector. Each row of the matrix equation represents an equation in the system. [I need an explanation, please!] Representing the system of equations as a matrix equation We are given the system of equations: $\begin{aligned}12x+15y+2z&=19 \\x-8y+6z&=14 \\15x+12y-6z&=3\end{aligned}$ First, let's rewrite this system to show the coefficients of each variable. $\begin{aligned}{12}x+{15}y+{2}z&=19 \\{1}x+({-8})y+{6}z&=14 \\{15}x+{12}y+({-6})z&=3\end{aligned}$ Now, the coefficient matrix can be written as follows. $\left[\begin{array} {ccc} {12} & {15} & {2} \\ {1} & {-8} & {6} \\ {15} & {12} & {-6} \end{array} \right]$ We can multiply this matrix by a column vector of variables and set it equal to a column vector with the values on the right side of the equations, as follows. $\left[\begin{array} {ccc} {12} & {15} & {2} \\ {1} & {-8} & {6} \\ {15} & {12} & {-6}\end{array} \right]\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right] =\left[\begin{array} {ccc} 19 \\ 14 \\ 3 \end{array} \right]$ This is our matrix equation. Summary $\text{A}$ and $\vec{b}$ are shown below. $\text{A}=\left[\begin{array} {ccc} 12 & 15 & 2 \\ 1 & -8 & 6 \\ 15 & 12 & -6 \end{array} \right]~~~~~~~~~~~~ \vec{b}=\left[\begin{array} {ccc} 19 \\ 14 \\ 3 \end{array} \right]$